Cohomological rigidity via toric degenerations

Jeudi 17 mai 14:00-15:00 - Milena Pabiniak - Cologne

Résumé : In order to study the homeomorphism type of manifolds, algebraic topology provides quite powerful invariants, as for example the integral cohomology ring. While this invariant does not distinguish (the homeomorphism type of) smooth manifolds, its restriction to certain natural classes of manifolds is known to be complete (for example to the class of simply connected closed 4-manifolds).
In this talk we will restrict our attention to a natural class of symplectic manifolds, called toric, which admit an action of a torus of large dimension, and we will pose a symplectic cohomological rigidity problem : is any ring isomorphism from the integral cohomology of M to that of N, which maps the class of a symplectic form of M to the class of a symplectic form of N, induced by a symplectomorphism ? Due to the symmetries coming from the torus action, toric symplectic manifolds are quite rigid, giving a hope for a positive answer to the above question.
To approach such a question one needs a tool for creating symplectomorphisms and I will explain how to use the construction of toric degenerations from algebraic geometry to that purpose. In particular, I will show that the cohomological rigidity problem holds for the family of Bott manifolds with rational cohomology ring isomorphic to that of a product of copies of CP^1. This is based on joint work with Sue Tolman.

Lieu : IMO, salle 2L8

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